Theorem fusgrvtxfi 29253

Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)

Hypothesis

Ref	Expression
isfusgr.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
fusgrvtxfi	⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Proof of Theorem fusgrvtxfi

Step	Hyp	Ref	Expression
1		isfusgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	isfusgr 29252	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3	2	simprbi 496	1 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6514  Fincfn 8921  Vtxcvtx 28930  USGraphcusgr 29083  FinUSGraphcfusgr 29250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-fusgr 29251

This theorem is referenced by:  fusgrfupgrfs  29265  nbfusgrlevtxm1  29311  nbfusgrlevtxm2  29312  nbusgrvtxm1  29313  uvtxnm1nbgr  29338  cusgrm1rusgr  29517  wlksnfi  29844  fusgrhashclwwlkn  30015  clwwlkndivn  30016  fusgreghash2wsp  30274  numclwwlk3lem2  30320  numclwwlk4  30322